Page 40 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

where T is a lower triangular matrix and D is a diagonal matrix. This relationship
has been used to develop rules for obtaining the inverse of A. A non-zero element of
the matrix T, say t , is the coefficient of relationship between animals i and j, if i and j
ij
are direct relatives or i = j and it is assumed that there is no inbreeding. Thus the
matrix T traces the flow of genes from one generation to the other; in other words,
it accounts only for direct (parent–offspring) relationships. It can easily be computed
applying the following rules.
For the ith animal:

t = 1
ii
If both parents (s and d) are known:

t = 0.5(t + t )
ij sj dj
If only one parent (s) is known:
t = 0.5(t )
ij sj
If neither parent is known:
t = 0
ij
The diagonal matrix D is the variance and covariance matrix for Mendelian
sampling. The Mendelian sampling (m) for an animal i with breeding value u and u
i s
and u as breeding values for its sire and dam, respectively, is:
d
m = u − 0.5(u + u ) (2.2)
i i s d
D has a simple structure and can easily be calculated. From Eqn 2.2, if both parents
of animal i are known, then:

var(m ) = var(u ) − var(0.5u + 0.5u )
i i s d
= var(u ) − var(0.5u ) − var(0.5u ) + 2cov(0.5u , 0.5u )
i s d s d
2 2 2 2
i u ss u dd u sd u
= (1 + F )s − 0.25a s − 0.25a s − 0.5a s
where a , a and a are elements of the relationship matrix A, and F is the inbreeding
ss dd sd i
coefficient of animal i.
2
var(m )/s = d = (1 + F ) − 0.25a − 0.25a − 0.5a
i u ii i ss dd sd
Since F = 0.5a
i sd
d = 1 − 0.25(1 + F ) − 0.25(1 + F )
ii s d
= 0.5 − 0.25(F + F )
s d
where F and F are the inbreeding coefficients of its sire and dam, respectively. If only
s d
one parent (s) is known, the diagonal element is:
d = 1 − 0.25(1 + F )
ii s
= 0.75 − 0.25(F )
s
and if no parent is known:
d = 1
ii

24 Chapter 2